Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}5x+2y &= -6 \\ -8x-2y &= -5\end{align*}$
Solution: Begin by moving the $x$ -term in the second equation to the right side of the equation. $-2y = 8x-5$ Divide both sides by $-2$ to isolate $y$ $y = {-4x + \dfrac{5}{2}}$ Substitute this expression for $y$ in the first equation. $5x+2({-4x + \dfrac{5}{2}}) = -6$ $5x - 8x + 5 = -6$ Simplify by combining terms, then solve for $x$ $-3x + 5 = -6$ $-3x = -11$ $x = \dfrac{11}{3}$ Substitute $\dfrac{11}{3}$ for $x$ back into the top equation. $5( \dfrac{11}{3})+2y = -6$ $\dfrac{55}{3}+2y = -6$ $2y = -\dfrac{73}{3}$ $y = -\dfrac{73}{6}$ The solution is $\enspace x = \dfrac{11}{3}, \enspace y = -\dfrac{73}{6}$.